Qualitative spatial representation and reasoning:
A hierarchical approach
2
Sanjiang Li1,2 and Bernhard Nebel2
1
Department of Computer Science and Technology
Tsinghua University, Beijing 100084, China
Institut für Informatik, Albert-Ludwigs-Universität Freiburg
D-79110 Freiburg, Germany
December 14, 2006
Abstract
The ability to reason in space is crucial for agents in order to make
informed decisions. Current high-level qualitative approaches to spatial
reasoning has serious deficiencies in not reflecting the hierarchical nature of spatial data and human spatial cognition. This paper proposes a
framework for hierarchical representation and reasoning about topological information, where a continuous model of space is approximated by
a collection of discrete sub-models, and spatial information is hierarchically represented in discrete sub-models in a rough set manner. The work
is based on the GRCC theory, where continuous and discrete models of
space are coped in a unified way. Reasoning issues such as determining
the mereological (part-whole) relations between two rough regions are also
discussed. Moreover, we consider an important problem that is closely related to map generalization in cartography and Geographical Information
Science. Given a spatial configuration at a finer level, we show how to construct a configuration at a coarser level while preserving the mereological
relations.
Keywords: Qualitative Spatial Reasoning; hierarchical spatial model;
Generalized Region Connection Calculus; resolution; map generalization
1
Introduction
The ability to reason in space is crucial for agents in order to make informed
decisions. There are in general two approaches for spatial representation and
reasoning. The low-level quantitative approach is based on Euclidean geometry and plays a predominate role in disciplines such as computational geometry
[17] and computer vision [11]. The second is the high-level qualitative approach,
known as Qualitative Spatial Reasoning (QSR). When describing spatial config-
1
urations, qualitative representation and reasoning can be worthwhile if precise,
quantitative information is not present or not desirable.
QSR is concerned with the qualitative aspects of representing and reasoning
about spatial entities. Among the many aspects of space, topology is perhaps
the most fundamental one. The Region Connection Calculus (RCC), initially
developed in [18, 19, 4] by the Leeds group, is widely recognized as the major
topological formalism for QSR.
1.1
The problem with current high-level approaches
Current high-level approaches to spatial reasoning, RCC in particular, suffer
from the following two serious deficiencies:
• they do not relate to quantitative spatial reasoning; and
• they do not reflect the hierarchical nature of spatial data and human
spatial cognition.
It has been widely recognized that the high-level, logic-based approaches
have little relevance to the quantitative approaches adopted in the acquisition,
storage, and manipulation of real-world data. One important reason for this lies
in the discrepancy between the continuous space models favored by high-level
approaches and the discrete, digital representations used at the low-level [9].
As a matter of fact, one axiom of RCC specifies that each region contains a
(non-tangential) proper part, which makes RCC has nothing to do with discrete
spaces (in the sense that each region is a union of atomic regions). On the
other hand, discrete spaces are evidently important in real-world applications
such as digital image processing, manipulation of various kinds of networks in
GIS, etc. This discrepancy between qualitative and quantitative approaches to
spatial information closely relates to the vector-raster debate in spatial data
handling [25].
The second problem is concerned with hierarchy and granularity. A granularity is formed by abstracting away from the world only related information, and
a series of abstracting activities result in a hierarchy of granularities. Hierarchy
is fundamental to human cognition [12, p.310], and “our ability to conceptualize
the world at different granularities and to switch among these granularities is
fundamental to our intelligence and flexibility.” [10]
In particular, since the spatial environment is infinitely complex, humans
typically use hierarchies as the major conceptual tool to structure and reason
about the infinite levels of details [30]. This coarse-to-fine approach is usually
very efficient since a lot of unrelated information has been discarded, and we
need to focus only on a rather restricted domain.
Current high-level formalisms of spatial reasoning, like RCC, consider only
ideal, infinite-precision information. It is strongly desirable to extend these
theories to support multi-representation and hierarchical reasoning.
2
The above two problems are closely related. On the one hand, to provide
a hierarchical approach to qualitative spatial reasoning, we need to relate discrete representations to the continuous models, which are favored by high-level
approaches. In particular we should develop a high-level approach to discrete
spaces that is compatible with existing formalisms of QSR. In this context, a
discrete model can be taken as an approximation of continuous ones at certain finite-precision. And, on the other hand, a hierarchical structure would
be crucial in understanding the relation between discrete and continuous representations of spatial information. In other words, the hierarchical method will
bridge the gap between high-level qualitative approaches to spatial information
and low-level quantitative ones.
1.2
Related work
In recent years efforts [8, 9, 15, 23] have been devoted to provide a high-level,
qualitative account for discrete spaces. These works, however, do not consider
problems like “How can a discrete model be linked to a continuous one?” and
“Can we construct continuous models step-by-step, using discrete (or even finite) models?” It is in this sense we say that the relationship between discrete
models and continuous ones is still unclear, and the connection between highlevel (qualitative) and low-level (quantitative) approaches to spatial information
handling is still missing.
The importance of hierarchical reasoning has been identified by several authors in the field of Geographical Information Science [30, 29, 2, 32]. Timpf
and Frank [30] give a definition of hierarchical spatial reasoning, using hierarchical spatial data structures, which computes increasingly better results in a
hierarchical fashion and stops the computation when a ‘good enough’ result is
achieved. Based on this general approach, Winter [32] proposes a hierarchical
method for determining mereological (part-whole) relations between two regions
represented as quadtrees [24].
Worboys [33] provides a formal framework for treating the notion of resolution and multi-resolution in geographic spaces. He goes further to develop
a rough set [16] like approach to reasoning with imprecision about spatial entities and relationships resulting from finite resolution representations. Stell
and Worboys [28] also introduce a concept of stratified map space to deal with
generalization and vagueness in multi-resolution spatial data handling.
Worboys’ approach is to a certain extent orthogonal to Timpf and Frank’s
hierarchical approach. On the one hand, the multi-resolution representation is
more general than the hierarchical representation in the sense that two resolutions may be incomparable, and on the other hand, Worboys does not consider
how to reason hierarchically in this framework.
1.3
Our approach
The main objective of this paper is to establish a hierarchical approach to highlevel spatial reasoning. Our research is based on the well known RCC theory,
3
which was first proposed by Randell, Cohn, and Cui [18, 19, 4], and studied by
many other researchers, see e.g. [21, 26, 6, 20, 13, 14]. In particular, Li and
Ying [14] propose a generalization of RCC, termed GRCC, which accommodates
both continuous (RCC) and discrete models of space. Several categorical notions such as sub-models and direct limits are introduced in the GRCC theory.
Moreover, an approach to constructing RCC models as direct limits of collections of finite models is also given. These notions can be useful for clarifying the
connection between discrete and continuous models, and it would be natural to
approximately represent space as a family of discrete models which varies over
a lattice of levels of details. This paper can be regarded as a continuation of
the research line started in Li and Ying [14].
We begin with two definitions of hierarchical spatial model in the GRCC
theory. Then we propose an approach to reasoning with imprecision about
spatial entities and relationships in a fixed resolution. We represent each object
x in a fixed resolution as a pair (ı(x), (x)) with the constraint that ı(x) is a
(possibly proper) part of (x). This rough set approach has been used by Cohn
and Gotts [3] and Worboys [33]. We further propose several methods to classify
the relationships between regions represented in a fixed resolution. Next, we give
rules to deduce the possible relation between two objects from the information
obtained at the present resolution. Moreover, we consider an important problem
that is closely related to map generalization in cartography and Geographical
Information Science. Given a spatial configuration at a finer level, we show how
to construct a configuration at a coarser level while preserving the mereological
relations.
1.4
Structure of this paper
The rest of this paper is organized as follows. In Section 2 we recall the generalized RCC theory proposed in [14] and then introduce two definitions of
hierarchical spatial models in Section 3. Section 4 concerns the representation
of regions and their relationships in a fixed resolution. In Section 5 we give
rules to determine the mereological relations between rough regions. Section 6
considers how to generalize information at finer resolution to coarser resolution without changing the relations between any two regions. Conclusions and
further work are given in Section 7.
2
Generalized Region Connection Calculus
In this section we recall some basic notions of the generalized RCC model
proposed in Li and Ying [14]. The Generalized Region Connection Calculus
(GRCC), following RCC, is an axiomatization of space that takes regions as
primitive.
Definition 2.1 (GRCC model [14]). A GRCC model is a complete Boolean
algebra B = hB; 0, 1, +, ·, −i together with a binary relation C on B such that
C, called a contact relation, satisfies the following conditions:
4
(C1) C is a reflexive and symmetric relation on B \ {0}.
(C2) For all x, y, z ∈ B \ {0, 1}, C(x + y, z) iff either C(x, z) or C(y, z) holds.
(C3) For all x ∈ B \ {0, 1}, C(x, −x), where −x is the complement of x in B.
Standard GRCC models arise from connected topological spaces.
Example 2.1. Suppose X is a connected topological space, write RC(X) for
the complete Boolean algebra of regular subsets of X. For any two nonempty
regular sets A, B ∈ RC(X), define ACX B if and only if A ∩ B 6= ∅. It is
straightforward to show that CX is a contact relation on RC(X). We call
hRC(X), CX i the standard GRCC model on X.
For a GRCC model hB, Ci, we often call a non-zero element in B a region,
and call 1 the universe. For two elements a, b in B, we write a ≤ b if a + b = b,
and write a < b if a ≤ b but not b ≤ a. An element a in B is an atom if b < a
implies b = 0 for all b ∈ B. Since elements in a GRCC model are interpreted as
regions, we also call ≤ the part-of relation, and write P for ≤.
In a GRCC model hB, Ci, using the part-of relation P and the contact
relation C, we can define a collection of relations on B. Table 1 summarizes the
definition of these relations. Note that the relations EQ,PO,O,DR,DC,EC
are symmetrical, and the relations P,PP,TPP,NTPP are non-symmetrical.
For a non-symmetrical relation R, we write R` for its converse. The relations
EQ, DR, PO, PP, PP`
(1)
form a jointly exhaustive and pairwise disjoint (JEPD) set of relations, which
are known as the RCC5 basic relations. The RCC5 algebra consists of unions
(or disjunctions) of RCC5 basic relations, and an RCC5 relation is also known
as a mereological relation or a part-whole relation.
Note that DR can be divided into EC and DC, PP (PP` , resp.) can be
divided into TPP and NTPP (TPP` and NTPP` , resp.). This results in
another JEPD set of relations, known as the RCC8 basic relations.
EQ, DC, EC, PO, TPP, TPP` , NTPP, NTPP`
(2)
The RCC8 algebra consists of unions (or disjunctions) of RCC8 basic relations,
and an RCC8 relation is also known as a topological relation.
Next, we consider two special kinds of GRCC models.
Definition 2.2 ([14]). A GRCC model hB, Ci is continuous if each region in B
contains a non-tangential proper part. We say hB, Ci is discrete if B is atomic
complete, i.e. each region in B is the sum of all atoms it contains.
A continuous GRCC model is precisely an RCC model [19]. Recently,
Düntsch and Winter [7] show that every continuous model is isomorphic to
a substructure of some standard model on a certain connected space X.
A discrete GRCC model corresponds to a model of Galton [9]. In fact, write
AT(B) for the set of atoms in B, and write A for the restriction of C to AT(B).
5
Relation
EQ
DR
PP
O
PO
DC
EC
TPP
NTPP
Table 1: Relations defined in GRCC
Interpretation
Definition
a is identical with b
a=b
a is discrete from b
a·b=0
a is a proper part of b
a<b
a overlaps b
a·b>0
a partially overlaps b
(aOb) ∧ (a 6≤ b) ∧ (a 6≥ b)
a is not in contact with b
¬(aCb)
a is in external contact with b
(aCb) ∧ (aDRb)
a is a tangential proper part of b
(aPPb) ∧ (aEC − b)
a is a non-tangential proper part of b
aDC − b
Then A is an adjacency relation in the sense of Galton [9]. The following lemma
links adjacency relations to contact relations.
Lemma 2.1 ([14]). Let B be a discrete GRCC model. Then two regions a, b in
B are in contact, i.e. aCb, if and only if we have two atoms w1 ≤ a and w2 ≤ b
such that w1 and w2 are in contact, i.e. w1 Aw2 .
3
Hierarchical spatial models: definitions and
examples
Intuitively, a hierarchical structure arises from a system of resolutions. In this
section we introduce two definitions of hierarchical models, and give two examples.
For a set of locations S, Worboys [33] defines a resolution to be a finite partition of S, where a partition is in the set-theoretical sense. This definition may
lead to some problems when we interpret elements of a partition as closed regions: adjacent elements share their boundary. This violates the set-theoretical
definition of a partition. In this paper we adopt the following lattice theoretical
definition of partition.
Definition 3.1 (Partition). Let B = hB; 0, 1, +, ·, −i be a complete Boolean
algebra. For a ∈ B, we say X = {di : i ∈ I} ⊂ B is a partition of a if the
following condition holds:
Σi∈I di = a and (∀i ∈ I)di 6= 0 and (∀i, j ∈ I)[i 6= j → di · dj = 0].
(3)
The following lemma characterizes the concept of partition in standard
GRCC models (see Example 2.1).
Lemma 3.1. Let X be a connected topological space, and let a be a nonempty
regular closed subset of X, i.e. a is a region in RC(X).
A collection of regions
S
{ai }i∈I in RC(X) is a partition of a if and only if i∈I ai = a, and for all i 6= j
we have (ai ∩ aj )◦ = ∅, where b and b◦ are, respectively, the closure and interior
of set b.
6
We now define the concept of resolution in the context of GRCC theory.
Definition 3.2 (Resolution). Given a GRCC model hB, Ci and a sub-algebra
B 0 of B, write C|B 0 for the restriction of C to B 0 × B 0 , and call hB 0 , C|B 0 i a
sub-model of hB, Ci. For a sub-model hB 0 , C|B 0 i, if B 0 is discrete, then we call
hB 0 , C|B 0 i a resolution of B, and call each atom of B 0 a B 0 -granule or a B 0 -pixel
of B.
We note here that if B 0 is a resolution of B, then the set of atomic regions
(i.e. granules) in B 0 is a partition of the universe 1.
For a GRCC model B, suppose R is the set of all resolutions of B. There
is a natural partial order on R. Given two resolutions B1 and B2 , we say B1
is finer than B2 , written B1 ≺ B2 , if B2 is a sub-algebra of B1 . In terms of
granules, B1 is finer than B2 if and only if each B1 -granule is contained in some
B2 -granule. In this sense, we say the grain size, or the granularity, of B1 is finer
than that of B2 .
Two resolutions B1 and B2 can be incomparable (cf. [33]). Moreover, we
have the following result.
Theorem 3.1. For a GRCC model B, suppose R is the set of all resolutions
of B. Set B1 t B2 = B1 ∩ B2 and set B1 u B2 to be the sub-algebra generated
by B1 ∪ B2 . Then hR; ≺; u, ti is a lattice.
Proof. This follows from that B1 tB2 (B1 uB2 , resp.) is the finest (the coarsest,
resp.) resolution which is coarser (finer, resp.) than both B1 and B2 .
We call hR; ≺; u, ti the resolution lattice of B. Note that this lattice may
be unbounded. On the one hand, we know 2 = {0, 1} is the coarsest resolution.
But on the other hand, there may be no finest resolution. This is because B
may not be discrete.
Given a GRCC model B, a hierarchical spatial model can be obtained by
associating a collection of resolutions of B, where each resolution can be taken
as an approximation of B at a certain coarse granularity.
Definition 3.3 (Hierarchical spatial model). Let B be a GRCC model, and let
R be its resolution lattice. A hierarchical spatial model over B is a pair (B, R1 ),
where R1 is a sub-lattice of R.
In GIS, hierarchical structures usually are obtained by recursively decomposing a plane region into sub-regions at different levels. This means, in our terms,
the set of resolutions we choose (viz. R1 ) is in liner order. This observation
leads to the following restricted version of a hierarchical model.
Definition 3.4 (Restricted hierarchical spatial model). Let B be an RCC
model, and let R be the resolution lattice of B. A restricted hierarchical spatial
model over B is a pair (B, R1 ), where
R1 = {B1 , B2 , · · · }
is a sequence
of resolutions in R such that Bi+1 strictly refines Bi for each i ≥ 1
S
and i≥1 Bi = B.
7
Y
6
p2
(0, 1)
(1, 1)
p3
p1 -
p4
(0, 0)
+
p5
-X
(1, 0)
Figure 1: A fragment of a regular hierarchical partition, where pi is an i-th
pixel, and pi+1 is a sub-pixel of p1 for 2 ≤ i ≤ 5.
For a restricted hierarchical spatial model B, each region in B can be completely determined in a certain fine enough resolution. This is because B is the
union of all resolutions Bi .
The most important example of hierarchical spatial model arises from hierarchical regular partitions of the real plane, where we hierarchically decompose
the plane into small squares of equal size, often called pixel s (see Figure 1).
This means each pixel in the i-th partition, called an i-pixel, can be further
decomposed into 4 equally sized pixels of the (i + 1)-th partition. A region
in the i-th partition is defined to be the union of a nonempty set of i-pixels.
Write Bi for the set of regions in the i-th partition together with the empty
set. Then each Bi is an atomic complete Boolean algebra, which is isomorphic
to the powerset algebra of the digital plane Z2 . Moreover, for each i ≥ 1, Bi
is a proper subalgebra of Bi+1 . Write BSfor the set of regions in all partitions
together with the empty set. Then B = i≥1 Bi and B is an atomless Boolean
algebra.
There are, among others, two possible ways of defining the contact relation
on B. They are based on, respectively, the well-known notion of 4- and 8neighborhood in digital topology [22] (see Figure 2). For a fixed i, two i-pixels
are 4-adjacent if they are either identical or 4-neighbors; correspondingly, two
regions a, b in Bi are in 4-contact if there are two 4-adjacent pixels contained
respectively in a and b.
Lemma 3.2. With the 4-contact relation each Bi is a discrete GRCC model.
Proof. Since each Bi is atomic complete, we need only to show that Bi satisfies
the three conditions (C1-3) in Definition 2.1. Take (C3) as an example. For a
region 0 6= x 6= 1 we show x is in 4-contact with its complement −x. There is
an i-pixel p contained in x such that x does not contain all 4-neighbors of p.
Write q for a 4-neighbor of p such that q is not contained in x. This means q is
contained in −x. Since p and q are 4-adjacent, we know x and its complement
are in 4-contact.
S
In general, for two regions a, b in B, recall that B = k≥1 Bk , there is some
i ≥ 1 such that a, b ∈ Bi . We define a, b to be in 4-contact if they are in 48
4
4 p 4
4
8 8 8
8 p 8
8 8 8
Figure 2: The 4 and 8 neighbors of a pixel p.
contact in Bi , i.e. there are two 4-adjacent i-pixels contained in a, b respectively.
This is well defined because, if two regions a, b are in 4-contact in Bi , then they
are also in 4-contact in Bj for any j ≥ i.
Write C4 for this contact relation on B. Then we have the following result.
S
Theorem 3.2. For B = k≥1 Bk and C4 defined as above, hB, C4 i is a continuous GRCC model.
Proof. By Lemma 3.2 it is clear that hB, C4 i is a GRCC model. To show it is
continuous, note that each i-pixel contains a (i + 2)-pixel as a non-tangential
proper part (ntpp for short). For example, p3 is an ntpp of p1 in Figure 1.
Similar definitions and results apply to hB, C8 i.
S
Theorem 3.3. For B = k≥1 Bk and C8 , hB, C8 i is a continuous GRCC
model.
It is straightforward to show that two regions in B are in 8-contact if and
only if they share a common point. Also note that each region in B is a plane
region. This suggests that hB, C8 i is a sub-model of the standard RCC model
on the plane (see Example 2.1).
But the situation is different for hB, C4 i. Two regions are in 4-contact if they
share a segment or a sub-region. Clearly, two 4-contact regions are necessarily
8-contact. The stronger 4-contact relation can be useful in describing situations,
say, a worm passing from one region to another without being perceived.
Remark 3.1. One reviewer raised the following interesting question: is it possible
each RCC model to be represented as a restricted hierarchical model? The
answer is negative, however. Take X = [0, 1]2 and consider the standard RCC
model RC(X). Suppose
S RC(X) can be represented as a restricted hierarchical
model by RC(X) = i≥1 Bi , where B1 , · · · , Bn , · · · is a sequence of resolutions
of RC(X) such that Bi+1 strictly refines Bi for all i ≥ 0. Note that RC(X) is a
continuous model. We can find a sequence p1 , p2 , · · · such that pk ∈ AT(Bik )
and pk+1 is an ntpp of pk for
T∞each k ≥ 1, where k ≤ ik < ik+1 for each
k ≥ 1. Let P be a point in k=1 pk . Such a point P exists because each pk
is a bounded closed, hence compact, subset of X. Let b be a square contained
in X such that P is one of its four endpoints. Clearly, we have b◦ ∩ p◦k 6= ∅
and pk 6⊆ b. Moreover, for each atomic region p in Bik , P is either an interior
or exteriorSpoint of p. This suggests that b is not a region
S in Bik . Therefore
S
RC(X) 6= k≥1 Bik . But by k ≤ ik and Bk ⊆ Bik we know k≥1 Bik = i≥1 Bi .
This is a contradiction.
9
a
a region a in B
a
lower appr.: ı(a)
a resolution B1 of B
upper appr.: (a)
Figure 3: The lower and upper approximation of a region a in the resolution B1
of a GRCC model B.
4
Spatial representation in a fixed resolution
In this section we discuss how spatial objects and their relation can be represented in a fixed resolution. In what follows, we suppose B is a GRCC model,
B1 is a nontrivial resolution of B, and write AT(B1 ) for its atoms set. This
means we require {0, 1} $ B1 $ B.
4.1
Rough regions
Since B1 is a proper subset of B, not all regions in B can have exact representations in B1 . We approximately represent all regions in B at the present
resolution in a rough set [16] manner. This method has been adopted by several
authors in QSR (see e.g. [3, 5, 33], but also see [1]).
Definition 4.1 (Lower and upper approximation). For a B-region x, we call
ı(x) ((x), resp.) the lower approximation (upper approximation, resp.) of x in
B1 , where
X
ı(x) =
{a ∈ B1 : a ≤ x},
(4)
Y
(x) =
{a ∈ B1 : a ≥ x}.
(5)
The above definition is well-defined since B1 , as a discrete sub-algebra of B,
is closed under sums and products.
For any B-region x, ı(x) is the largest B1 -region contained in x and (x) is
the smallest B1 -region containing x. (see Figure 3) We can interpret ı(x) as the
B1 -interior of x, and (x) as the B1 -closure of x. We stress that ı(x) could be
0.
Since B1 is generated by its atoms set AT(B1 ), we have the following results.
Lemma 4.1. For any B-region x, we have
X
ı(x) =
{w ∈ AT(B1 ) : w ≤ x},
X
(x) =
{w ∈ AT(B1 ) : w · x 6= 0}.
10
(6)
(7)
In this paper we represent each region in x ∈ B as a pair (ı(x), (x)) at the
present resolution B1 , and call x a rough region in B1 . If x happens to be in
B1 , we say x is a crisp region in B1 , and otherwise call x a vague region in B1 .
Note that for a region x in B, x ∈ B1 if and only if ı(x) = (x).
Remark 4.1. Our approach is similar to that of Worboys [33] in the way spatial objects are represented in a fixed resolution. But Worboys [33] allows
more relaxed representations, where a region x may be represented as any pair
(l(x), u(x)) with the constraint l(x) ≤ ı(x) ≤ (x) ≤ u(x). The egg-yolk approach [3], however, is developed directly on a continuous model and does not
assume a discrete resolution.
4.2
Rough relations
The RCC5 mereological relations and the RCC8 topological relations are designed for crisp regions. In this section we describe ways of extending these
relations to rough regions in a fixed resolution.
Suppose B is a GRCC model and B1 is a nontrivial resolution of B, i.e.
{0, 1} ⊂ B1 ⊂ B. For a JEPD set of relations B on crisp regions, we now extend
relations in B to rough regions.
In the remainder of this paper, we require the lower approximation ı(x) of a
rough region x = (ı(x), (x)) in B1 to be nonzero. This is natural since, given a
fixed resolution, we usually are only interested in objects that can be precisely
represented to a certain extent at the present resolution.
Definition 4.2 (Rough relation). Given two regions x, y ∈ B, the rough relation
R between x, y is represented as a 4-tuple
hρ(ı(x), ı(y)), ρ((x), (y)), ρ(ı(x), (y)), ρ((x), ı(y))i
(8)
where ρ(u, v) is the B-relation between crisp regions u and v in B1 .
Remark 4.2. There are some other methods for approximately representing relations between rough regions, where some instead of all crisp relations in the
above 4-tuple are considered. The lower approximation method (LAM) considers only the relation between ı(x) and ı(y), while the upper approximation
method (UAM) considers the relation between (x) and (y). These two methods are rather imprecise. The lower-upper approximation method (LUAM)
combines the previous two methods and consider both the relation between ı(x)
and ı(y) and the relation between (x) and (y).
We fix some notations here. In what follows, we denote rough relations by
capital roman fonts R, S, T (and some times M, N ), and use corresponding bold
fonts, often with subscripts, say Rı , to denote the crisp relations between their
approximations. For a rough relation R, we write, respectively, Rıı , R , Rı , Rı
for the B-relations between (ı(x), ı(y)), ((x), (y)), (ı(x), (y)), and ((x), ı(y)).
This means R = hRıı , R , Rı , Rı i.
Since each of Rıı , R , Rı , Rı can be any basic relation in B, there are
|B|4 many possible rough relations between rough regions. A question that
11
arises naturally is to determine whether a 4-tuple R is realizable, i.e. there
are two rough regions related by R. When B contains a large set of relations,
the computation work is arduous and error-prone. For example, for the rather
small RCC8 basic relations, we should check 84 = 4096 4-tuples. Fortunately,
questions like this can be reformulated as a constraint satisfaction problem.
We take the RCC5 relations as an example, but the approach can be extended to other RCC systems. Our approach is based on the following theorem,
which reduces the computation of realizable 4-tuples to a simple consistency
checking work.
Theorem 4.1. An RCC5 4-tuple hRıı , R , Rı , Rı i is a possible rough relation
between two rough regions if and only if the following RCC5 constraint network
Θ is consistent.
Θ = {ı(x)P(x), ı(y)P(y), ı(x)Rıı ı(y), (x)R (y), ı(x)Rı (y), (x)Rı ı(y)}
where Rıı , R , Rı , Rı are all RCC5 basic relations.
Proof. Note that relations appeared in Θ are in the set
S = {P, EQ, DR, PO, PP, PP` }.
c8 , one maximal tractable fragment of RCC8 identified
This set is contained in H
by Renz and Nebel [21]. Now since path-consistency decides consistency for any
c8 , we can apply a path-consistency algorithm for each
constraint network over H
RCC5 4-tuple to decide whether it is a realizable rough relation.
There are 54 = 625 RCC5 4-tuples. But only 51 survive the above test.
These 51 rough relations are those (46 cases) given in Table 7 and the following
• hEQ, EQ, EQ, EQi (if x and y are two identical crisp regions),
• hEQ, PP, PP, EQi (if x is crisp and y is vague, and x = ı(y)),
• hPP` , EQ, EQ, PP` i (if x is crisp and y is vague, and x = (y)),
• hEQ, PP` , EQ, PP` i (the converse of the second case),
• hPP, EQ, PP, EQi (the converse of third case).
Remark 4.3. Similar classifications have been made by Cohn and Gotts [3] and
Stell [27] for egg-yolk regions, where an egg-yolk region in a continuous model B
is a pair (r, s) such that r, s ∈ R and r ≤ s. Assuming r 6= s and r, s > 0, Cohn
and Gotts find 46 possible relations between egg-yolk regions, which are showed
in Table 7. On the other hand, Stell shows that there are 85 realizable relations
if 0 ≤ r ≤ s. Note that r and s may be any region in B. This is different from
our approach: We explicitly require a fixed resolution B1 and only consider
egg-yolk regions in B1 instead of B. Moreover, in our approach each egg-yolk
region in B1 is an approximation of some region in B. Furthermore, we use
Theorem 4.1 to determine whether a 4-tuple is realizable. This is different from
the approaches of Cohn and Gotts [3] and Stell [27].
12
Table 2: Relations between a crisp region and a vague region
(ı(x), ı(y)) = ((x), ı(y)) ((x), (y)) = (ı(x), (y)) (x, y)
EQ
PP
PP
DR
DR
DR
DR
PP
PO
PO
PO
DR
PP
PP
PP
PP`
EQ
PP`
PP`
PP
PO
PP`
PP`
PP`
PP`
PO
PO
PO
PP
PO
PO
PO
PO
Quite often it is desirable to make further classifications according to whether
a and/or b is crisp in B1 . There are 4 cases. Recall that we assume ı(x) > 0
and ı(y) > 0.
c-c Both x and y are crisp regions. In this case, ρ(ı(x), ı(y)) = ρ((x), (y)) =
ρ(ı(x), (y)) = ρ((x), ı(y)), there are only 5 possible relations, viz. the 5
basic RCC5 relations.
c-v x is crisp and y is vague. In this case, ρ(ı(x), ı(y)) = ρ((x), ı(y)) and
ρ((x), (y)) = ρ(ı(x), (y)), there are 11 possible relations, which are
showed in Table 2.
v-c x is vague and y is crisp. In this case, ρ(ı(x), ı(y)) = ρ(ı(x), (y)) and
ρ((x), (y)) = ρ((x), ı(y)), there are also 11 possible relations, which are
showed in Table 3.
v-v Both x and y are vague regions. This case corresponds to the egg-yolk
calculus by Cohn and Gotts [3]. There are altogether 46 possible relations
in this case (see Table 7).
Remark 4.4. Suppose we adopt the lower-upper approximation method (LUAM),
where the relation between x and y is approximately represented by ρ(ı(x), ı(y))
and ρ((x), (y)). There are at most 25 possible different relations. But since
ı(x) ≤ (x) and ı(y) ≤ (y), the relation pairs (R, DR) are impossible, where
R can be either of EQ, PO, PP, PP` . So using LUAM we obtain 21 possible
relations. These are summarized in Table 6.
5
Reasoning in a fixed resolution
In this section we consider how to reason about information represented in a
fixed resolution.
13
Table 3: Relations between a vague region and a crisp region
(ı(x), ı(y)) = (ı(x), (y)) ((x), (y)) = ((x), ı(y)) (x, y)
EQ
PP`
PP`
DR
DR
DR
DR
PP`
PO
DR
PO
PO
EQ
PP
PP
PP
PP
PP
PP
PP`
PO
PO
PO
PP
`
`
PP
PP
PP`
PO
PP`
PO
PO
PO
PO
Suppose B is a GRCC model and B1 is a resolution of B. Given two rough
regions x, y in B1 , suppose we know (or partially know) the rough mereological
relation between x and y. Then what about the RCC5 relation between x and
y? For example, if we know ı(x)DRı(y) and (x) = (y), then what can be said
about the RCC5 relation between x and y. Can they be discrete? Equal?
In what follows, we restrict our discussion to rough regions that have nonzero
lower approximation. We have the following basic rules:
(1) If x ≤ y, then ı(x) ≤ ı(y) and (x) ≤ (y);
(2) If x ≥ y, then ı(x) ≥ ı(y) and (x) ≥ (y).
More rules can be obtained by applying the basic rules together with the
assumptions that ı(x) > 0 and ı(y) > 0. Two examples are given below.
Lemma 5.1. If ı(x)DRı(y), then xDRy or xPOy.
Proof. By the basic rules we know x 6≤ y and x 6≥ y. The conclusion follows
directly.
Under the assumption that each rough region has a nonzero lower approximation, we have the following lemma.
Lemma 5.2. Suppose (x) = (y). Then x overlaps y.
Proof. This is because by ı(x) > 0 we have an atomic region w ∈ B1 such that
w ≤ ı(x). Note that w is also contained in (y) since (x) = (y). By Equations 6
and 7, we have w ≤ x and w · y > 0. Therefore x · y > 0, hence xOy.
We summarize these rules in the following tables. In what follows, we say
an RCC5 relation R is definite if it is a basic relation, and say R is indefinite
if it is not.
Table 4 asserts which RCC5 relation can hold between x and y if we know
the definite RCC5 relation between ı(x) and ı(y). Table 5 infers the RCC5
14
Table 4: Determining the RCC5 relations using lower approximation
(ı(x), ı(y)) EQ
DR
PP
PP`
PO
(x, y)
O
PO,DR PO,PP PO,PP` PO
Table 5: Determining the RCC5 relations using upper approximation
((x), (y)) EQ DR
PP
PP`
PO
`
(x, y)
O
DR PO,PP PO,PP
PO,DR
relation between x and y given the definite RCC5 relation between (x) and
(y). Table 6 combines the results of Table 4 and Table 5, where there are 8
(out of 21) situations that cannot lead to a definite RCC5 relation.
To obtain more precise information, we need to consider all relations in the
4-tuple hRıı , R , Rı , Rı i. Recall in the last section we have divided the 51
(realizable) rough mereological relations into 4 groups according to whether a
and/or b is crisp. There are altogether 73 cases: 5 for crisp-crisp regions, 11 for
crisp-vague, and 11 for vague-crisp, and 46 for vague-vague. If a, b are both crisp,
then Rıı = R = Rı = Rı and the RCC5 relation between a, b is definite.
Interestingly, if only one of a, b is crisp, then the RCC5 relation between a, b is
also definite, see Table 2 and Table 3. As for the 46 vague-vague rough relations,
Table 7 shows that 10 situations can lead to indefinite RCC5 relations.
From Tables 2, 3, 7, we can see that altogether nine RCC5 relations can hold
between two rough regions. These are
EQ, DR, PP, PP` , PO, O, {PO, PP}, {PO, PP` }, {PO, DR}
(9)
There are 5 definite and 4 indefinite RCC5 relations. It is natural to classify the
73 rough relations into 9 groups, each of which leads to a (possibly indefinite)
RCC5 relation.
Note that a row in Table 7 can be explained as a rule. Take the first
row as example. For any two regions x, y ∈ B, suppose ρ(ı(x), ı(y)) = EQ,
ρ((x), (y)) = EQ, ρ(ı(x), (y)) = PP, and ρ((x), ı(y)) = PP` . Then Table 7 asserts that the RCC5 relation between x, y is O, which is an indefinite RCC5 relation. We also say that O is determined by the rough rela-
Table 6: Determining the RCC5 relations using lower-upper approximation
((x), (y))
EQ
DR
PP
PP`
PO
EQ
O
PO,PP
PO,PP`
PO
DR
PO
DR
PO
PO
DR,PO
PP
PP,PO
PP,PO
PO
PO
(ı(x), ı(y))
PP` PP` ,PO
PO
PO,PP`
PO
PO
PO
PO
PO
PO
15
Table 7: Determining the RCC5 relations using rough relations
(ı(x), ı(y))
EQ
EQ
EQ
EQ
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
DR
PP
PP
PP
PP
PP
PP
PP
PP
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PO
PO
PO
PO
PO
PO
PO
PO
PO
((x), (y))
EQ
PO
PP
PP`
EQ
PP
PP
PP
PP`
PP`
PP`
PO
PO
PO
PO
PO
PO
PO
PO
PO
DR
PO
PO
PP
PP
PP
PP
PP`
EQ
PO
PO
PP`
PP`
PP`
PP`
PP
EQ
PO
PO
PO
PO
PP
PP
PP`
PP`
EQ
(ı(x), (y))
PP
PP
PP
PP
PP
PP
PP
PP
DR
PO
PP
DR
DR
DR
PO
PO
PO
PP
PP
PP
DR
PP
PP
PP
PP
PP
PP
PP
PP
PO
PP
PP`
PO
EQ
PP
PP
PP
PO
PO
PP
PP
PP
PP
PO
PP
PP
16
((x), ı(y))
PP`
PP`
PP`
PP`
PP`
DR
PO
PP`
PP`
PP`
PP`
DR
PO
PP`
DR
PO
PP`
DR
PO
PP`
DR
PO
PP`
PP
PO
EQ
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PP`
PO
PP`
PO
PP`
PO
PP`
PP`
PP`
PP`
(x, y)
O
PO
PO,PP
PO,PP`
PO
PO
PO
PO
PO
PO
PO
DR,PO
PO
PO
PO
PO
PO
PO
PO
PO
DR
PO
PO
PP
PO,PP
PP
PO,PP
PO
PO,PP
PO
PO
PP`
PP` ,PO
PP`
PP` ,PO
PO
PP` ,PO
PO
PO
PO
PO
PO
PO
PO
PO
PO
tion hEQ, EQ, PP, PP` i. Another example. Row 12 of Table 7 asserts that
{PO, DR} is determined by the rough relation hDR, PO, DR, DRi.
Definition 5.1. Let B1 be a resolution of a GRCC model B, and R an RCC5
relation. For two regions a, b in B, we say (the fact) aRb can be determined
at resolution B1 if and only if R can be determined by the rough relation
hρ(ı(a), ı(b)), ρ((a), (b)), ρ(ı(a), (b)), ρ((a), ı(b))i, where ρ(x, y) is the RCC5
basic relation between B1 regions x, y.
Given two rough regions a = (ı(a), (a)), b = (ı(b), (b)), the rough information are often not sufficient to decide the definite RCC5 relation between
a, b. For example, suppose we know that ı(a) = ı(b) < (a) = (b), and this is
the only information we have about a, b at the present resolution. Then we do
not know whether a is a proper part of b—they may also be equal or partially
overlap. But if we luckily know that (a) ≤ ı(b) and ı(a) < (b), then we are
sure that a is a proper part of b. Moreover, if either (a) 6≤ ı(b) or ı(a) 6< (b),
then there are two possibilities: (i) a is not a proper part of b; or (ii) a may be
(but we are not sure) a proper part of b. In other words, if a is really a proper
part of b, then this fact can be ascertained at the current resolution if and only
if we know (a) ≤ ı(b) and ı(a) < (b).
The following theorem summarizes when a definite RCC5 relation can be
determined.
Theorem 5.1. Let B be a GRCC model and B1 be a non-trivial resolution of
B. For two rough regions a, b in B1 , we have
1. The fact aEQb can be determined at resolution B1 if and only if a, b are
identical crisp regions in B1 ;
2. The fact aDRb can be determined at resolution B1 if and only if (a)DR(b);
3. The fact aPPb can be determined at resolution B1 if and only if (a) ≤ ı(b)
and ı(a) < (b);
4. The fact aPP` b can be determined at resolution B1 if and only if ı(a) ≥
(b) and (a) > ı(b);
5. The fact aPOb can be determined at resolution B1 if and only if the following conditions are satisfied:
– ı(a) 6≤ ı(b) or (a) 6≤ (b); and
– ı(a) 6≥ ı(b) or (a) 6≥ (b); and
– ı(a) · (b) > 0 or (a) · ı(b) > 0.
6
Generalization of a spatial configuration
In the above sections we considered ways to represent and reason with spatial
information in a fixed resolution, where a region is represented as a pair in a
17
resolution, and relations between rough regions are characterized by the 4-tuples
of relations between their approximations.
In this section we consider, given a spatial configuration of n objects in a
higher resolution, how to find another configuration of these objects in a coarser
resolution while preserving the mereological relations. By ‘spatial configuration’
we mean a network of (crisp) regions in a spatial model, say the real plane or one
of its resolution, where (mereological) relations among regions can be explicitly
determined.
This question relates closely to map generalization in cartography and Geographic Information Science [31], where the objective of generalization is to
produce maps at coarser levels of detail without changing essential characteristics of underlying geographic information.
We now give a detailed description of the question. Suppose B is a continuous model and, B1 , S
B2 , · · · , Bi , · · · is a collection of finer and finer resolutions
of B such that B = i≥1 Bi . This means B is a restricted hierarchical spatial
model (see Definition 3.4).
Given a spatial configuration Ω = {x1 , x2 , · · · , xn } in B, suppose Bλ is the
first resolution of B in which all xi ∈ Bλ . Therefore, all xi can be crisply represented in Bλ , but not all can be crisply represented in any coarser resolution.
The question now is: “For a coarser resolution Bh , can we find a reasonable approximation x∗i ∈ Bh for each i without changing their mereological relations?”
By a reasonable approximation we mean x∗i should be between the lower and
upper approximation of xi in Bh . Of course, the smaller h is the better.
Definition 6.1. For a spatial configuration Ω = {x1 , x2 , · · · , xn } in B, write
λ = λ(Ω) for the lowest level at which all xi are crisp. We call a set of n
crisp regions Ω∗ = {x∗1 , x∗2 , · · · , x∗n } at level h ≤ λ a generalization of Ω if the
following conditions are satisfied:
(1) The RCC5 relation between x∗i and x∗j is the same as that between xi and
xj (1 ≤ i, j ≤ n).
(2) ıh (xi ) ≤ x∗i ≤ h (xi ) (1 ≤ i ≤ n), where ıh (xi ) and h (xi ) are, respectively,
the lower and upper approximation of xi at level h.
For a spatial configuration Ω, set κ(Ω) to be the lowest level at which all
mereological relations between objects in Ω can be determined. We first note
that there may be no generalizations for Ω at any level l ≤ κ(Ω).
Indeed, consider the hierarchical spatial model (B, {Bl }l≥1 ) that is generated
by hierarchical regular partitions of the unit square, as show in Figure 4, where
B0 = {0, 1} is the coarsest level of B, B1 , B2 , B3 are the first three levels of B,
and pi (1 ≤ i ≤ 4) is a pixel at level 1, and pij (1 ≤ i, j ≤ 4) is a pixel at level
2, and pijk (1 ≤ i, j, k ≤ 4) is a pixel at level 3.
Now set
x = p1 + p412 + p413 + p421 + p424
y = p4 + p13 + p142
z = p1 + p4 + p23 + p24 + p311 + p312 + p321 + p322
18
(10)
(11)
(12)
p11 p12 p21 p22
p1
p2
p14 p13 p24 p23
p41 p42 p31 p32
p4
B0
p3
p44 p43 p34 p33
B1
B2
B3
Figure 4: A hierarchical spatial model and its first 3 levels.
See Figure 5 for illustrations. Consider the spatial configuration Ω = {x, y, z}.
Clearly, x, y, z are vague at levels 1 and 2, and crisp at level 3. Note that
xPOy, xPPz, and yPPz. These relations can be determined at level 1, namely
κ(Ω) = 1. This is because, at level 1,
(ı1 (x), 1 (x)) = (p1 , p1 + p4 )
(ı1 (y), 1 (y)) = (p4 , p1 + p4 )
(ı1 (z), 1 (z)) = (p1 + p2 , p1 + p2 + p3 + p4 )
where we add to operators ı and  a subscript, here 1, to indicate the level at
which the region is lower or upper approximated.
We claim that Ω has no generalization at level κ(Ω) = 1. To show this,
suppose {x∗ , y ∗ , z ∗ } is a generalization of Ω at level 1. Then
p1 = ı1 (x) ≤ x∗
p4 = ı1 (y) ≤ y ∗
p1 + p4 = ı1 (z) ≤ z ∗
≤ 1 (x) = p1 + p4
≤ 1 (y) = p1 + p4
≤ 1 (z) = p1 + p2 + p3 + p4
Since x∗ , y ∗ , z ∗ are crisp regions in B1 , x∗ and y ∗ must be either p1 or p1 + p4 .
But if x∗ = p1 , then x∗ ≤ y ∗ ; if x∗ = p1 + p4 , then x∗ ≥ y ∗ . Both cases
contradict x∗ POy ∗ , which follows from the assumption that {x∗ , y ∗ , z ∗ } is a
generalization of Ω.
Although generalizations cannot be found for all spatial configuration Ω at
level κ(Ω), we can certainly find a generalization at level κ(Ω) + 1. To prove this
result, we need the following lemma, where two regions x, y are k-equivalent,
written x ∼k y, if they have the same lower and upper approximation at level
k, i.e. ık (x) = ık (y) and k (x) = k (y).
Lemma 6.1. Suppose (B, {Bl }l≥1 ) is a restricted hierarchical spatial model,
where each pixel at level l contains more than one sub-pixels at level l + 1.
Then, for any region x ∈ B and for any resolution Bk , we can find a crisp
region x∗ in Bk+1 such that x∗ ∼k x.
Proof. To construct a region x∗ which is k-equivalent to x, our idea is, roughly
speaking, reserving all (k+1)-pixels that are contained in x, and deleting at least
19
x
x
x
x
x xx
x xx
x xx
x xx
xx
xx
y
y
y
y
x
y y
y
y y
y y
y y
y y
y
y
y
y
y
y
y
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
z
zz z z z
zz z z z
zz z z z
z
z
z
z
Figure 5: A spatial configuration Ω = {x, y, z}.
one (k + 1)-pixel from a k-pixel that is at the ‘boundary’ of x. More precisely,
we choose for each k-pixel w a set, written Xw , of its sub-pixels at level k + 1.
Write S(w) for the set of sub-pixels of w at level k + 1. Then
• if w ≤ x, set Xw = S(w);
• if w · x = 0, set Xw = ∅;
• if 0 < w · x < w, set Xw to be a nonempty proper subset of S(w) which
contains all sub-pixels w0 that is contained in x.
The last case is possible since there is more than one sub-pixel in S(w) and not
all sub-pixels are contained in x. S
Now sum up all sub-pixels in {Xw : w is a k-pixel}. The result x∗ is a
crisp region in Bk+1 which is k-equivalent to x.
Also note that each x∗ constructed as above is also between the lower and
upper approximations of x at level k + 1, i.e. ık+1 (x) ≤ x∗ ≤ k+1 (x). We now
arrive at the main result of this section:
Theorem 6.1. Suppose (B, {Bl }l≥1 ) is a restricted hierarchical spatial model,
where each pixel at level l contains more than one sub-pixel at level l + 1. Then
for any spatial configuration Ω, we have a generalization of Ω at any level k ≥
κ(Ω) + 1, where κ(Ω) is the coarsest level at which relations among regions in
Ω can be determined.
Proof. Suppose Ω = {x1 , x2 , · · · , xn }. For each level k ≥ κ(Ω) + 1, we now
construct a realization of Ω at level k. For each xi , by Lemma 6.1, we can find
a crisp region x∗i at level k which is (k − 1)-equivalent to xi . We claim that
{x∗1 , x∗2 , · · · , x∗n } is a realization of Ω at level k. To show this, we only need to
prove that the RCC5 relation between x∗i and x∗j is the same as that between xi
and xj . Recall that the relation of xi and xj can be determined at level κ(Ω).
Now, since the rough relation of x∗i and x∗j at level k is the same as that of xi
and xj , the relation between x∗i and x∗j can also be determined at level κ(Ω),
which can only be the relation of xi and xj .
20
Obviously, computing the answer is of lower polynomial complexity.
Now return to the example described in Figure 5. Set x∗ = p1 + p41 , y ∗ =
p4 + p13 , z ∗ = p1 + p4 + p23 + p24 + p31 . Then {x∗ , y ∗ , z ∗ } is a generalization of
Ω at level 2 = κ(Ω) + 1.
7
Conclusions and further work
In this paper we have proposed a framework for hierarchical representation and
reasoning about topological information. In our hierarchical spatial model, each
region is represented as rough regions at various levels of details. Mereological
relations between rough regions are characterized by the RCC5 relation between
lower and upper approximations. We gave rules (Table 7) for determining the
RCC5 relation between two rough regions using simply the rough relation between the rough regions. We also gave a method for constructing mereological
generalizations of spatial configuration at coarser levels of details.
Although only RCC5 relations were discussed in this paper, our approach
also applies to RCC8 and other RCC systems of relations. We take the view
that, for rough regions, mereological relations are more important than finer
topological distinctions. This is partially because rough RCC8 relations rarely
provide information for us to differ TPP from NTPP, and to differ from EC
from DC, where a rough RCC8 relation between two rough regions a, b is a 4tuple hρ(ı(a), ı(b)), ρ((a), (b)), ρ(ı(a), (b)), ρ((a), ı(b))i with ρ(x, y) the RCC8
basic relation between x and y. In our opinion, it is not necessary to make finer
topological distinctions until we know the definite mereological relation.
There are several directions for further work.
An important reasoning problem we have not discussed is the problem of
deciding the consistency of a rough relation network. The solution to this problem can be based either on the RCC5 composition table or on a rough RCC5
composition table.
Winter [32] proposes a method for hierarchically determining the mereological relation between two quadtree regions. It is our future work to extend this
and other hierarchical reasoning techniques to general hierarchical models.
Acknowledgements
This work was partially supported by the Alexander von Humboldt Foundation,
the NSFC (60673105, 60321002, 60496321), a DFG grant (No. 623/1-3), and the
DFG project Logospace as part of the collaborative research center SGB/TR-8
“Spatial Cognition”. We also gratefully acknowledge the invaluable suggestions
of three anonymous referees.
21
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